• Previous Article
    Measure-preservation criteria for a certain class of 1-lipschitz functions on Zp in mahler's expansion
  • DCDS Home
  • This Issue
  • Next Article
    Metastable energy strata in numerical discretizations of weakly nonlinear wave equations
August  2017, 37(7): 3749-3786. doi: 10.3934/dcds.2017159

Blow-up solutions for two coupled Gross-Pitaevskii equations with attractive interactions

1. 

Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, P.O. Box 71010, Wuhan 430071, China

2. 

Department of Mathematics, School of Science, Wuhan University of Technology, Wuhan 430070, China

* Corresponding author

Received  August 2016 Revised  March 2017 Published  April 2017

Fund Project: This work was supported by NSFC (grants 11322104 and 11671394 for YJGuo, grant 11501555 for XYZeng and grant 11471331 for HSZhou)

The paper is concerned with a system of two coupled time-independent Gross-Pitaevskii equations in $\mathbb{R}^2$, which is used to model two-component Bose-Einstein condensates with both attractive intraspecies and attractive interspecies interactions. This system is essentially an eigenvalue problem of a stationary nonlinear Schrödinger system in $\mathbb{R}^2$, and solutions of the problem are obtained by seeking minimizers of the associated variational functional with constrained mass (i.e. $L^2-$norm constaints). Under a certain type of trapping potentials $V_i(x)$ ($i=1, 2$), the existence, non-existence and uniqueness of this kind of solutions are studied. Moreover, by establishing some delicate energy estimates, we show that each component of the solutions blows up at the same point (i.e., one of the global minima of $V_i(x)$) when the total interaction strength of intraspecies and interspecies goes to a critical value. An optimal blowing up rate for the solutions of the system is also given.

Citation: Yujin Guo, Xiaoyu Zeng, Huan-Song Zhou. Blow-up solutions for two coupled Gross-Pitaevskii equations with attractive interactions. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3749-3786. doi: 10.3934/dcds.2017159
References:
[1]

M. H. AndersonJ. R. EnsherM. R. MatthewsC. E. Wieman and E. A. Cornell, Observation of Bose-Einstein condensation in a dilute atomic vapor, Collected Papers of Carl Wieman, (2008), 453-456.  doi: 10.1142/9789812813787_0062.  Google Scholar

[2]

W. H. AschbacherJ. FröhlichG. M. GrafK. Schnee and M. Troyer, Symmetry breaking regime in the nonlinear Hartree equation, J. Math. Phys., 43 (2002), 3879-3891.  doi: 10.1063/1.1488673.  Google Scholar

[3]

W. Z. Bao and Y. Y. Cai, Ground states of two-component Bose-Einstein condensates with an internal atomic Josephson junction, East Asian J. Appl. Math., 1 (2011), 49-81.  doi: 10.4208/eajam.190310.170510a.  Google Scholar

[4]

W. Z. Bao and Y. Y. Cai, Mathematical theory and numerical methods for Bose-Einstein condensation, Kinetic and Related Models, 6 (2013), 1-135.  doi: 10.3934/krm.2013.6.1.  Google Scholar

[5]

T. BartschL. Jeanjean and N. Soave, Normalized solutions for a system of coupled cubic Schrödinger equations on $\mathbb{R}^3$, J. Math. Pures Appl., 106 (2016), 583-614.  doi: 10.1016/j.matpur.2016.03.004.  Google Scholar

[6]

T. Bartsch and Z.-Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on $\mathbb{R}^N$, Comm. Partial Differential Equations, 20 (1995), 1725-1741.  doi: 10.1080/03605309508821149.  Google Scholar

[7]

M. Caliari and M. Squassina, Location and phase segregation of ground and excited states for 2D Gross-Pitaevskii systems, Dynamics of PDE, 5 (2008), 117-137.  doi: 10.4310/DPDE.2008.v5.n2.a2.  Google Scholar

[8]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics 10, Courant Institute of Mathematical Science/AMS, New York, 2003. doi: 10.1090/cln/010.  Google Scholar

[9]

Z. J. Chen and W. M. Zou, Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent, Arch. Ration. Mech Anal., 205 (2012), 515-551.  doi: 10.1007/s00205-012-0513-8.  Google Scholar

[10]

Z. J. Chen and W. M. Zou, Standing waves for coupled Schrödinger equations with decaying potentials, J. Math. Phys., 54 (2013), 111505, 21pp.  doi: 10.1063/1.4833795.  Google Scholar

[11]

Z. J. Chen and W. M. Zou, An optimal constant for the existence of least energy solutions of a coupled Schrödinger system, Calc. Var. Partial Differential Equations, 48 (2013), 695-711.  doi: 10.1007/s00526-012-0568-2.  Google Scholar

[12]

F. DalfovoS. GiorginiL. P. Pitaevskii and S. Stringari, Theory of Bose-Einstein condensation in trapped gases, Rev. Modern Phys., 71 (1999), 463-512.   Google Scholar

[13] B. GidasW. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbb{R}^N$, in Mathematical analysis and applications Part A, Adv. in Math. Suppl. Stud., Academic Press, New York, 1981.   Google Scholar
[14]

Y. J. Guo and R. Seiringer, On the mass concentration for Bose-Einstein condensates with attractive interactions, Lett. Math. Phys., 104 (2014), 141-156.  doi: 10.1007/s11005-013-0667-9.  Google Scholar

[15]

Y. J. Guo, Z. -Q. Wang, X. Y. Zeng and H. -S. Zhou, Properties of ground states of attractive Gross-Pitaevskii equations with multi-well potentials, preprint, arXiv: 1502.01839. Google Scholar

[16]

Y. J. GuoX.Y. Zeng and H.-S. Zhou, Energy estimates and symmetry breaking in attractive Bose-Einstein condensates with ring-shaped potentials, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 809-828.  doi: 10.1016/j.anihpc.2015.01.005.  Google Scholar

[17]

Y. J. Guo, X. Y. Zeng and H. -S. Zhou, Blow-up solutions for a nonlinear Schrödinger system with attractive intraspecies and repulsive interspecies interactions, preprint. Google Scholar

[18]

D. S. HallM. R. MatthewsJ. R. EnsherC. E. Wieman and E. A. Cornell, Dynamics of component separation in a binary mixture of Bose-Einstein condensates, Collected Papers of Carl Wieman, (2008), 515-518.  doi: 10.1142/9789812813787_0071.  Google Scholar

[19]

N. Ikoma and K. Tanaka, A local mountain pass type result for a system of nonlinear Schrödinger equations, Calc. Var. Partial Differential Equations, 40 (2011), 449-480.  doi: 10.1007/s00526-010-0347-x.  Google Scholar

[20]

R. K. Jackson and M. I. Weinstein, Geometric analysis of bifurcation and symmetry breaking in a Gross-Pitaevskii equation, J. Stat. Phys., 116 (2004), 881-905.  doi: 10.1023/B:JOSS.0000037238.94034.75.  Google Scholar

[21]

O. Kavian and F. B. Weissler, Self-similar solutions of the pseudo-conformally invariant nonlinear Schrödinger equation, Michigan Math. J., 41 (1994), 151-173.  doi: 10.1307/mmj/1029004922.  Google Scholar

[22]

E. W. KirrP. G. Kevrekidis and D. E. Pelinovsky, Symmetry-breaking bifurcation in the nonlinear Schrödinger equation with symmetric potentials, Comm. Math. Phys., 308 (2011), 795-844.  doi: 10.1007/s00220-011-1361-3.  Google Scholar

[23]

E. W. KirrP. G. KevrekidisE. Shlizerman and M. I. Weinstein, Symmetry-breaking bifurcation in nonlinear Schrödinger/Gross-Pitaevskii equations, SIAM J. Math. Anal., 40 (2008), 566-604.  doi: 10.1137/060678427.  Google Scholar

[24]

Y. C. KuoW. W. Lin and S. F. Shieh, Bifurcation analysis of a two-component Bose-Einstein condensate, Phys. D, 211 (2005), 311-346.  doi: 10.1016/j.physd.2005.09.003.  Google Scholar

[25]

T. C. Lin and J. C. Wei, Spikes in two-component systems of nonlinear Schrödinger equations with trapping potentials, J. Differential Equations, 229 (2006), 538-569.  doi: 10.1016/j.jde.2005.12.011.  Google Scholar

[26]

P. L. Lions, The concentration-compactness principle in the Caclulus of Variations. The locally compact case, Part Ⅰ: Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109-145. Part Ⅱ: Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 223-283. doi: 10. 1016/S0294-1449(16)30422-X.  Google Scholar

[27]

C. Y. LiuN.V. Nguyen and Z.-Q. Wang, Orbital stability of spatially synchronized solitary waves of an m-coupled nonlinear Schrödinger system, J. Math. Phys., 57 (2016), 101501, 20pp.  doi: 10.1063/1.4964255.  Google Scholar

[28]

Z.L. Liu and Z.-Q. Wang, Ground states and bound states of a nonlinear Schrödinger system, Advanced Nonlinear Studies, 10 (2010), 175-193.  doi: 10.1515/ans-2010-0109.  Google Scholar

[29]

M. Maeda, On the symmetry of the ground states of nonlinear Schrödinger equation with potential, Adv. Nonlinear Stud., 10 (2010), 895-925.  doi: 10.1515/ans-2010-0409.  Google Scholar

[30]

E. MontefuscoB. Pellacci and M. Squassina, Semiclassical states for weakly coupled nonlinear Schrödinger systems, J. Eur. Math. Soc., 10 (2008), 47-71.  doi: 10.4171/JEMS/103.  Google Scholar

[31]

N.V. Nguyen and Z.-Q. Wang, Orbital stability of solitary waves for a nonlinear Schrödinger system, Adv. Differ. Equations, 16 (2011), 977-1000.   Google Scholar

[32]

W.-M. Ni and I. Takagi, On the shape of least-energy solutions to a semilinear Neumann problem, Comm. Pure Appl. Math., 44 (1991), 819-851.  doi: 10.1002/cpa.3160440705.  Google Scholar

[33]

A. Pomponio, Coupled nonlinear Schrödinger systems with potentials, J. Differ. Equations, 227 (2006), 258-281.  doi: 10.1016/j.jde.2005.09.002.  Google Scholar

[34] M. Reed and B. Simon, Methods of Modern Mathematical Physics. ô. Analysis of Operators, Academic Press,, New York-London, 1978.   Google Scholar
[35]

J. Royo-Letelier, Segregation and symmetry breaking of strongly coupled two-component Bose-Einstein condensates in a harmonic trap, Calc. Var. Partial Differential Equations, 49 (2014), 103-124.  doi: 10.1007/s00526-012-0571-7.  Google Scholar

[36]

W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.  doi: 10.1007/BF01626517.  Google Scholar

[37]

M. Struwe, Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Ergebnisse Math. 34, Springer, 2008.  Google Scholar

[38]

E. Timmermans, Phase separation of Bose-Einstein condensates, Phys. Rev. Lett., 81 (1998), 5718-5721.  doi: 10.1103/PhysRevLett.81.5718.  Google Scholar

[39]

X. F. Wang, On concentration of positive bound states of nonlinear Schrödinger equations, Comm. Math. Phys., 153 (1993), 229-244.  doi: 10.1007/BF02096642.  Google Scholar

[40]

J. Wei and T. Weth, Radial solutions and phase separation in a system of two coupled schrödinger equations, Arch. Rational. Mech. Anal., 190 (2008), 83-106.  doi: 10.1007/s00205-008-0121-9.  Google Scholar

[41]

M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolations estimates, Comm. Math. Phys., 87 (1983), 567-576.   Google Scholar

[42]

X. Y. ZengY.M. Zhang and H.-S. Zhou, Existence and stability of standing waves for a coupled nonlinear schrödinger system, Acta Mathematica Scientia, 35 (2015), 45-70.  doi: 10.1016/S0252-9602(14)60138-7.  Google Scholar

show all references

References:
[1]

M. H. AndersonJ. R. EnsherM. R. MatthewsC. E. Wieman and E. A. Cornell, Observation of Bose-Einstein condensation in a dilute atomic vapor, Collected Papers of Carl Wieman, (2008), 453-456.  doi: 10.1142/9789812813787_0062.  Google Scholar

[2]

W. H. AschbacherJ. FröhlichG. M. GrafK. Schnee and M. Troyer, Symmetry breaking regime in the nonlinear Hartree equation, J. Math. Phys., 43 (2002), 3879-3891.  doi: 10.1063/1.1488673.  Google Scholar

[3]

W. Z. Bao and Y. Y. Cai, Ground states of two-component Bose-Einstein condensates with an internal atomic Josephson junction, East Asian J. Appl. Math., 1 (2011), 49-81.  doi: 10.4208/eajam.190310.170510a.  Google Scholar

[4]

W. Z. Bao and Y. Y. Cai, Mathematical theory and numerical methods for Bose-Einstein condensation, Kinetic and Related Models, 6 (2013), 1-135.  doi: 10.3934/krm.2013.6.1.  Google Scholar

[5]

T. BartschL. Jeanjean and N. Soave, Normalized solutions for a system of coupled cubic Schrödinger equations on $\mathbb{R}^3$, J. Math. Pures Appl., 106 (2016), 583-614.  doi: 10.1016/j.matpur.2016.03.004.  Google Scholar

[6]

T. Bartsch and Z.-Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on $\mathbb{R}^N$, Comm. Partial Differential Equations, 20 (1995), 1725-1741.  doi: 10.1080/03605309508821149.  Google Scholar

[7]

M. Caliari and M. Squassina, Location and phase segregation of ground and excited states for 2D Gross-Pitaevskii systems, Dynamics of PDE, 5 (2008), 117-137.  doi: 10.4310/DPDE.2008.v5.n2.a2.  Google Scholar

[8]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics 10, Courant Institute of Mathematical Science/AMS, New York, 2003. doi: 10.1090/cln/010.  Google Scholar

[9]

Z. J. Chen and W. M. Zou, Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent, Arch. Ration. Mech Anal., 205 (2012), 515-551.  doi: 10.1007/s00205-012-0513-8.  Google Scholar

[10]

Z. J. Chen and W. M. Zou, Standing waves for coupled Schrödinger equations with decaying potentials, J. Math. Phys., 54 (2013), 111505, 21pp.  doi: 10.1063/1.4833795.  Google Scholar

[11]

Z. J. Chen and W. M. Zou, An optimal constant for the existence of least energy solutions of a coupled Schrödinger system, Calc. Var. Partial Differential Equations, 48 (2013), 695-711.  doi: 10.1007/s00526-012-0568-2.  Google Scholar

[12]

F. DalfovoS. GiorginiL. P. Pitaevskii and S. Stringari, Theory of Bose-Einstein condensation in trapped gases, Rev. Modern Phys., 71 (1999), 463-512.   Google Scholar

[13] B. GidasW. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbb{R}^N$, in Mathematical analysis and applications Part A, Adv. in Math. Suppl. Stud., Academic Press, New York, 1981.   Google Scholar
[14]

Y. J. Guo and R. Seiringer, On the mass concentration for Bose-Einstein condensates with attractive interactions, Lett. Math. Phys., 104 (2014), 141-156.  doi: 10.1007/s11005-013-0667-9.  Google Scholar

[15]

Y. J. Guo, Z. -Q. Wang, X. Y. Zeng and H. -S. Zhou, Properties of ground states of attractive Gross-Pitaevskii equations with multi-well potentials, preprint, arXiv: 1502.01839. Google Scholar

[16]

Y. J. GuoX.Y. Zeng and H.-S. Zhou, Energy estimates and symmetry breaking in attractive Bose-Einstein condensates with ring-shaped potentials, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 809-828.  doi: 10.1016/j.anihpc.2015.01.005.  Google Scholar

[17]

Y. J. Guo, X. Y. Zeng and H. -S. Zhou, Blow-up solutions for a nonlinear Schrödinger system with attractive intraspecies and repulsive interspecies interactions, preprint. Google Scholar

[18]

D. S. HallM. R. MatthewsJ. R. EnsherC. E. Wieman and E. A. Cornell, Dynamics of component separation in a binary mixture of Bose-Einstein condensates, Collected Papers of Carl Wieman, (2008), 515-518.  doi: 10.1142/9789812813787_0071.  Google Scholar

[19]

N. Ikoma and K. Tanaka, A local mountain pass type result for a system of nonlinear Schrödinger equations, Calc. Var. Partial Differential Equations, 40 (2011), 449-480.  doi: 10.1007/s00526-010-0347-x.  Google Scholar

[20]

R. K. Jackson and M. I. Weinstein, Geometric analysis of bifurcation and symmetry breaking in a Gross-Pitaevskii equation, J. Stat. Phys., 116 (2004), 881-905.  doi: 10.1023/B:JOSS.0000037238.94034.75.  Google Scholar

[21]

O. Kavian and F. B. Weissler, Self-similar solutions of the pseudo-conformally invariant nonlinear Schrödinger equation, Michigan Math. J., 41 (1994), 151-173.  doi: 10.1307/mmj/1029004922.  Google Scholar

[22]

E. W. KirrP. G. Kevrekidis and D. E. Pelinovsky, Symmetry-breaking bifurcation in the nonlinear Schrödinger equation with symmetric potentials, Comm. Math. Phys., 308 (2011), 795-844.  doi: 10.1007/s00220-011-1361-3.  Google Scholar

[23]

E. W. KirrP. G. KevrekidisE. Shlizerman and M. I. Weinstein, Symmetry-breaking bifurcation in nonlinear Schrödinger/Gross-Pitaevskii equations, SIAM J. Math. Anal., 40 (2008), 566-604.  doi: 10.1137/060678427.  Google Scholar

[24]

Y. C. KuoW. W. Lin and S. F. Shieh, Bifurcation analysis of a two-component Bose-Einstein condensate, Phys. D, 211 (2005), 311-346.  doi: 10.1016/j.physd.2005.09.003.  Google Scholar

[25]

T. C. Lin and J. C. Wei, Spikes in two-component systems of nonlinear Schrödinger equations with trapping potentials, J. Differential Equations, 229 (2006), 538-569.  doi: 10.1016/j.jde.2005.12.011.  Google Scholar

[26]

P. L. Lions, The concentration-compactness principle in the Caclulus of Variations. The locally compact case, Part Ⅰ: Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109-145. Part Ⅱ: Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 223-283. doi: 10. 1016/S0294-1449(16)30422-X.  Google Scholar

[27]

C. Y. LiuN.V. Nguyen and Z.-Q. Wang, Orbital stability of spatially synchronized solitary waves of an m-coupled nonlinear Schrödinger system, J. Math. Phys., 57 (2016), 101501, 20pp.  doi: 10.1063/1.4964255.  Google Scholar

[28]

Z.L. Liu and Z.-Q. Wang, Ground states and bound states of a nonlinear Schrödinger system, Advanced Nonlinear Studies, 10 (2010), 175-193.  doi: 10.1515/ans-2010-0109.  Google Scholar

[29]

M. Maeda, On the symmetry of the ground states of nonlinear Schrödinger equation with potential, Adv. Nonlinear Stud., 10 (2010), 895-925.  doi: 10.1515/ans-2010-0409.  Google Scholar

[30]

E. MontefuscoB. Pellacci and M. Squassina, Semiclassical states for weakly coupled nonlinear Schrödinger systems, J. Eur. Math. Soc., 10 (2008), 47-71.  doi: 10.4171/JEMS/103.  Google Scholar

[31]

N.V. Nguyen and Z.-Q. Wang, Orbital stability of solitary waves for a nonlinear Schrödinger system, Adv. Differ. Equations, 16 (2011), 977-1000.   Google Scholar

[32]

W.-M. Ni and I. Takagi, On the shape of least-energy solutions to a semilinear Neumann problem, Comm. Pure Appl. Math., 44 (1991), 819-851.  doi: 10.1002/cpa.3160440705.  Google Scholar

[33]

A. Pomponio, Coupled nonlinear Schrödinger systems with potentials, J. Differ. Equations, 227 (2006), 258-281.  doi: 10.1016/j.jde.2005.09.002.  Google Scholar

[34] M. Reed and B. Simon, Methods of Modern Mathematical Physics. ô. Analysis of Operators, Academic Press,, New York-London, 1978.   Google Scholar
[35]

J. Royo-Letelier, Segregation and symmetry breaking of strongly coupled two-component Bose-Einstein condensates in a harmonic trap, Calc. Var. Partial Differential Equations, 49 (2014), 103-124.  doi: 10.1007/s00526-012-0571-7.  Google Scholar

[36]

W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.  doi: 10.1007/BF01626517.  Google Scholar

[37]

M. Struwe, Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Ergebnisse Math. 34, Springer, 2008.  Google Scholar

[38]

E. Timmermans, Phase separation of Bose-Einstein condensates, Phys. Rev. Lett., 81 (1998), 5718-5721.  doi: 10.1103/PhysRevLett.81.5718.  Google Scholar

[39]

X. F. Wang, On concentration of positive bound states of nonlinear Schrödinger equations, Comm. Math. Phys., 153 (1993), 229-244.  doi: 10.1007/BF02096642.  Google Scholar

[40]

J. Wei and T. Weth, Radial solutions and phase separation in a system of two coupled schrödinger equations, Arch. Rational. Mech. Anal., 190 (2008), 83-106.  doi: 10.1007/s00205-008-0121-9.  Google Scholar

[41]

M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolations estimates, Comm. Math. Phys., 87 (1983), 567-576.   Google Scholar

[42]

X. Y. ZengY.M. Zhang and H.-S. Zhou, Existence and stability of standing waves for a coupled nonlinear schrödinger system, Acta Mathematica Scientia, 35 (2015), 45-70.  doi: 10.1016/S0252-9602(14)60138-7.  Google Scholar

[1]

Roy H. Goodman, Jeremy L. Marzuola, Michael I. Weinstein. Self-trapping and Josephson tunneling solutions to the nonlinear Schrödinger / Gross-Pitaevskii equation. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 225-246. doi: 10.3934/dcds.2015.35.225

[2]

Jeremy L. Marzuola, Michael I. Weinstein. Long time dynamics near the symmetry breaking bifurcation for nonlinear Schrödinger/Gross-Pitaevskii equations. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1505-1554. doi: 10.3934/dcds.2010.28.1505

[3]

Xiaoyu Zeng, Yimin Zhang. Asymptotic behaviors of ground states for a modified Gross-Pitaevskii equation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5263-5273. doi: 10.3934/dcds.2019214

[4]

Patrick Henning, Johan Wärnegård. Numerical comparison of mass-conservative schemes for the Gross-Pitaevskii equation. Kinetic & Related Models, 2019, 12 (6) : 1247-1271. doi: 10.3934/krm.2019048

[5]

Norman E. Dancer. On the converse problem for the Gross-Pitaevskii equations with a large parameter. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : 2481-2493. doi: 10.3934/dcds.2014.34.2481

[6]

Georgy L. Alfimov, Pavel P. Kizin, Dmitry A. Zezyulin. Gap solitons for the repulsive Gross-Pitaevskii equation with periodic potential: Coding and method for computation. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1207-1229. doi: 10.3934/dcdsb.2017059

[7]

Dong Deng, Ruikuan Liu. Bifurcation solutions of Gross-Pitaevskii equations for spin-1 Bose-Einstein condensates. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3175-3193. doi: 10.3934/dcdsb.2018306

[8]

Ko-Shin Chen, Peter Sternberg. Dynamics of Ginzburg-Landau and Gross-Pitaevskii vortices on manifolds. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 1905-1931. doi: 10.3934/dcds.2014.34.1905

[9]

Thomas Chen, Nataša Pavlović. On the Cauchy problem for focusing and defocusing Gross-Pitaevskii hierarchies. Discrete & Continuous Dynamical Systems - A, 2010, 27 (2) : 715-739. doi: 10.3934/dcds.2010.27.715

[10]

E. Norman Dancer. On a degree associated with the Gross-Pitaevskii system with a large parameter. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 1835-1839. doi: 10.3934/dcdss.2019120

[11]

Dapeng Du, Yifei Wu, Kaijun Zhang. On blow-up criterion for the nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3639-3650. doi: 10.3934/dcds.2016.36.3639

[12]

Laurent Di Menza, Olivier Goubet. Stabilizing blow up solutions to nonlinear schrÖdinger equations. Communications on Pure & Applied Analysis, 2017, 16 (3) : 1059-1082. doi: 10.3934/cpaa.2017051

[13]

Türker Özsarı. Blow-up of solutions of nonlinear Schrödinger equations with oscillating nonlinearities. Communications on Pure & Applied Analysis, 2019, 18 (1) : 539-558. doi: 10.3934/cpaa.2019027

[14]

Shuai Li, Jingjing Yan, Xincai Zhu. Constraint minimizers of perturbed gross-pitaevskii energy functionals in $\mathbb{R}^N$. Communications on Pure & Applied Analysis, 2019, 18 (1) : 65-81. doi: 10.3934/cpaa.2019005

[15]

Jian Zhang, Shihui Zhu, Xiaoguang Li. Rate of $L^2$-concentration of the blow-up solution for critical nonlinear Schrödinger equation with potential. Mathematical Control & Related Fields, 2011, 1 (1) : 119-127. doi: 10.3934/mcrf.2011.1.119

[16]

Binhua Feng. On the blow-up solutions for the fractional nonlinear Schrödinger equation with combined power-type nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1785-1804. doi: 10.3934/cpaa.2018085

[17]

Hristo Genev, George Venkov. Soliton and blow-up solutions to the time-dependent Schrödinger-Hartree equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (5) : 903-923. doi: 10.3934/dcdss.2012.5.903

[18]

Van Duong Dinh. On blow-up solutions to the focusing mass-critical nonlinear fractional Schrödinger equation. Communications on Pure & Applied Analysis, 2019, 18 (2) : 689-708. doi: 10.3934/cpaa.2019034

[19]

Jianbo Cui, Jialin Hong, Liying Sun. On global existence and blow-up for damped stochastic nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6837-6854. doi: 10.3934/dcdsb.2019169

[20]

Cristophe Besse, Rémi Carles, Norbert J. Mauser, Hans Peter Stimming. Monotonicity properties of the blow-up time for nonlinear Schrödinger equations: Numerical evidence. Discrete & Continuous Dynamical Systems - B, 2008, 9 (1) : 11-36. doi: 10.3934/dcdsb.2008.9.11

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (46)
  • HTML views (21)
  • Cited by (3)

Other articles
by authors

[Back to Top]